# Why are break-even and shut down points based on average cost?

I understand the basic idea of break-even and shut-down points, where break-even is the price at which revenue covers all economic costs, and is located where the marginal cost equals to average cost. And that shut down is the price where revenue just covers variable costs (it is where marginal cost equals average variable cost).

My question, why are taking in to account average cost and average variable cost, rather than just total cost and just variable cost? Why is shut down point where revenue equals average variable cost rather than just variable cost or total cost, and why is break even where marginal cost equals average total cost, not just total cost or variable cost?

Thank you

• It seems to be the point where revenue per widget equals cost per widget i.e. where total revenue equals total cost. – user253751 Feb 10 '20 at 14:12

When you are finding the break even point, you are looking for a price-point.

The price is essentially the average (and marginal) revenue. Suppose price of good $$x$$ is $$p$$.

Revenue = $$p * x$$.

Average Revenue = Revenue / $$x$$ = $$p$$.

So what you are really doing when looking for p is to set Average Revenue = Average Cost.

At that point, you will also have the fact that Revenue = Cost. So your logic carries through. You still break even at the point where revenues exactly cover costs.

To see this, note Average Costs = Costs / $$x$$.

So Average Costs = Average Revenue $$\iff$$ Costs / $$x$$ = Revenue / $$x$$ $$\iff$$ Costs = Revenue

The interpretation when focusing on average costs is only slightly different. When talking about averages, we're just talking about prices and costs per unit. So you break even at the point where your revenue per unit equals your cost per unit. It is necessarily implied that you also break even where your total revenue equals your total cost.

• Thanks a lot, so for the last paragraph, where you talk about revenue per unit equals cost per unit, is cost per unit the marginal cost? Some resources say where marginal cost equals average cost, others say where average cost equals average revenue, which one is it? Don't we need to take into account margin cost? – Christopher U Feb 10 '20 at 3:27
• I'm guessing MC = ATC = P for break-even point? If so, is there a particular reason why MC = ATC at that point – Christopher U Feb 10 '20 at 3:28
• Well MC=p for a price taking firm in perfect competition. Also MC=AC always represents the minimum point of AC. – BB King Feb 10 '20 at 3:35
• Of course we mean price of good x at quantity x. But at this level of instruction the OP is at prices are almost always taken as given and constant when discussing break even for the first time. – BB King Feb 10 '20 at 14:51
• MC=AC at minimum of AC and P=MC always are two facts that are true in your setting (price taking firms in perfect competition) independently of each other. The combination of these two facts answers your question. As to why these two facts are true in the first place, that would be a different question. A nice implication here is that the break even point is truly the minimum price necessary, as that’s as low as AC can go, but that’s only an interpretation not a necessary condition. – BB King Feb 10 '20 at 18:18

My students also frequently make the mistake of thinking they "understand the basic idea of" something, when what they actually know is a formula to calculate some aspect of it.

Average costs being equal to marginal costs in the break even point is not the basic idea, it is a mathematical property that only holds given certain assumptions (e.g., price taker firm).

The actual basic idea is that in the break even point price is so low that the best a price taker firm can do is to act in a manner where its revenues will cover its costs and break even. From this it follows that revenue per unit (i.e. price) is equal to the minimum of average costs, it is impossible to further decrease cost per unit by scaling up (or down) production. From profit maximization and price taking it also follows that at this quantity price also equals marginal costs, and by transitivity you have $$AC = MC$$, but again, this is a mathematical property, not the idea.

The basic idea of shut down points is that the price is so low that companies can no longer break even. However, in some cases it is possible that producing positive quantities is still beneficial as sunk costs are not recoverable anyway. This occurs when the price is higher than the minimum of average variable costs. The logic is the same as in the previous case, but sunk costs are not considered, as those are sunk anyway.